/***************************************************************************
  **************************************************************************
  
                Spherical Harmonic Transform Kit 2.7
  
  
   Contact: Peter Kostelec
            geelong@cs.dartmouth.edu
  
  
   Copyright 1997-2003  Sean Moore, Dennis Healy,
                        Dan Rockmore, Peter Kostelec
  
  
   Copyright 2004  Peter Kostelec, Dan Rockmore


     SpharmonicKit is free software; you can redistribute it and/or modify
     it under the terms of the GNU General Public License as published by
     the Free Software Foundation; either version 2 of the License, or
     (at your option) any later version.
  
     SpharmonicKit is distributed in the hope that it will be useful,
     but WITHOUT ANY WARRANTY; without even the implied warranty of
     MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the
     GNU General Public License for more details.
  
     You should have received a copy of the GNU General Public License
     along with this program; if not, write to the Free Software
     Foundation, Inc., 675 Mass Ave, Cambridge, MA 02139, USA.
  
  
   Commercial use is absolutely prohibited.
  
   See the accompanying LICENSE file for details.
  
  ************************************************************************
  ************************************************************************/


/* these are the bit-reverse, 4096 roots of unity, and by subrecursion,
   contains all Nth roots of unity (N < 4096) in bit-reversed order */


const double r4096[8192] =
{1.000000000000000, 0, -1.000000000000000, 0, 0, 
 1.000000000000000, 0, -1.000000000000000, 
  0.707106781186548, 0.707106781186547, -0.707106781186547, 
  -0.707106781186548, -0.707106781186547, 0.707106781186548, 
  0.707106781186548, -0.707106781186547, 0.923879532511287, 
  0.382683432365090, -0.923879532511287, -0.382683432365090, 
  -0.382683432365090, 0.923879532511287, 0.382683432365090, 
  -0.923879532511287, 0.382683432365090, 0.923879532511287, 
  -0.382683432365090, -0.923879532511287, -0.923879532511287, 
  0.382683432365090, 0.923879532511287, -0.382683432365090, 
  0.980785280403230, 0.1950903220161282, -0.980785280403230, 
  -0.1950903220161286, -0.1950903220161282, 0.980785280403230, 
  0.1950903220161283, -0.980785280403230, 0.555570233019602, 
  0.831469612302545, -0.555570233019602, -0.831469612302545, 
  -0.831469612302545, 0.555570233019602, 0.831469612302545, 
  -0.555570233019602, 0.831469612302545, 0.555570233019602, 
  -0.831469612302545, -0.555570233019602, -0.555570233019602, 
  0.831469612302545, 0.555570233019602, -0.831469612302545, 
  0.1950903220161283, 0.980785280403230, -0.1950903220161282, 
  -0.980785280403230, -0.980785280403230, 0.1950903220161286, 
  0.980785280403230, -0.1950903220161282, 0.995184726672197, 
  0.0980171403295606, -0.995184726672197, -0.0980171403295608, 
  -0.0980171403295607, 0.995184726672197, 0.0980171403295608, 
  -0.995184726672197, 0.634393284163646, 0.773010453362737, 
  -0.634393284163645, -0.773010453362737, -0.773010453362737, 
  0.634393284163645, 0.773010453362737, -0.634393284163645, 
  0.881921264348355, 0.471396736825998, -0.881921264348355, 
  -0.471396736825998, -0.471396736825998, 0.881921264348355, 
  0.471396736825998, -0.881921264348355, 0.290284677254462, 
  0.956940335732209, -0.290284677254462, -0.956940335732209, 
  -0.956940335732209, 0.290284677254462, 0.956940335732209, 
  -0.290284677254462, 0.956940335732209, 0.290284677254462, 
  -0.956940335732209, -0.290284677254462, -0.290284677254462, 
  0.956940335732209, 0.290284677254462, -0.956940335732209, 
  0.471396736825998, 0.881921264348355, -0.471396736825998, 
  -0.881921264348355, -0.881921264348355, 0.471396736825998, 
  0.881921264348355, -0.471396736825998, 0.773010453362737, 
  0.634393284163645, -0.773010453362737, -0.634393284163645, 
  -0.634393284163645, 0.773010453362737, 0.634393284163646, 
  -0.773010453362737, 0.0980171403295608, 0.995184726672197, 
  -0.0980171403295607, -0.995184726672197, -0.995184726672197, 
  0.0980171403295608, 0.995184726672197, -0.0980171403295606, 
  0.998795456205172, 0.04906767432741802, -0.998795456205172, 
  -0.04906767432741798, -0.04906767432741801, 0.998795456205172, 
  0.04906767432741814, -0.998795456205172, 0.671558954847018, 
  0.740951125354959, -0.671558954847018, -0.740951125354959, 
  -0.740951125354959, 0.671558954847019, 0.740951125354959, 
  -0.671558954847018, 0.903989293123443, 0.427555093430282, 
  -0.903989293123443, -0.427555093430282, -0.427555093430282, 
  0.903989293123443, 0.427555093430282, -0.903989293123443, 
  0.336889853392220, 0.941544065183021, -0.336889853392220, 
  -0.941544065183021, -0.941544065183021, 0.336889853392220, 
  0.941544065183021, -0.336889853392220, 0.970031253194544, 
  0.2429801799032639, -0.970031253194544, -0.242980179903264, 
  -0.2429801799032639, 0.970031253194544, 0.242980179903264, 
  -0.970031253194544, 0.514102744193222, 0.857728610000272, 
  -0.514102744193222, -0.857728610000272, -0.857728610000272, 
  0.514102744193222, 0.857728610000272, -0.514102744193222, 
  0.803207531480645, 0.595699304492433, -0.803207531480645, 
  -0.595699304492433, -0.595699304492433, 0.803207531480645, 
  0.595699304492433, -0.803207531480645, 0.1467304744553617, 
  0.989176509964781, -0.1467304744553616, -0.989176509964781, 
  -0.989176509964781, 0.1467304744553618, 0.989176509964781, 
  -0.1467304744553617, 0.989176509964781, 0.1467304744553617, 
  -0.989176509964781, -0.1467304744553618, -0.1467304744553616, 
  0.989176509964781, 0.1467304744553617, -0.989176509964781, 
  0.595699304492433, 0.803207531480645, -0.595699304492433, 
  -0.803207531480645, -0.803207531480645, 0.595699304492433, 
  0.803207531480645, -0.595699304492433, 0.857728610000272, 
  0.514102744193222, -0.857728610000272, -0.514102744193222, 
  -0.514102744193222, 0.857728610000272, 0.514102744193222, 
  -0.857728610000272, 0.242980179903264, 0.970031253194544, 
  -0.2429801799032639, -0.970031253194544, -0.970031253194544, 
  0.242980179903264, 0.970031253194544, -0.2429801799032639, 
  0.941544065183021, 0.336889853392220, -0.941544065183021, 
  -0.336889853392220, -0.336889853392220, 0.941544065183021, 
  0.336889853392220, -0.941544065183021, 0.427555093430282, 
  0.903989293123443, -0.427555093430282, -0.903989293123443, 
  -0.903989293123443, 0.427555093430282, 0.903989293123443, 
  -0.427555093430282, 0.740951125354959, 0.671558954847018, 
  -0.740951125354959, -0.671558954847019, -0.671558954847018, 
  0.740951125354959, 0.671558954847018, -0.740951125354959, 
  0.04906767432741814, 0.998795456205172, -0.04906767432741801, 
  -0.998795456205172, -0.998795456205172, 0.04906767432741798, 
  0.998795456205172, -0.04906767432741802, 0.999698818696204, 
  0.02454122852291229, -0.999698818696204, -0.02454122852291233, 
  -0.02454122852291214, 0.999698818696204, 0.02454122852291227, 
  -0.999698818696204, 0.689540544737067, 0.724247082951467, 
  -0.689540544737067, -0.724247082951467, -0.724247082951467, 
  0.689540544737067, 0.724247082951467, -0.689540544737067, 
  0.914209755703531, 0.405241314004990, -0.914209755703531, 
  -0.405241314004990, -0.405241314004990, 0.914209755703531, 
  0.405241314004990, -0.914209755703531, 0.359895036534988, 
  0.932992798834739, -0.359895036534988, -0.932992798834739, 
  -0.932992798834739, 0.359895036534988, 0.932992798834739, 
  -0.359895036534988, 0.975702130038529, 0.2191012401568698, 
  -0.975702130038528, -0.21910124015687, -0.2191012401568697, 
  0.975702130038529, 0.2191012401568698, -0.975702130038529, 
  0.534997619887097, 0.844853565249707, -0.534997619887097, 
  -0.844853565249707, -0.844853565249707, 0.534997619887097, 
  0.844853565249707, -0.534997619887097, 0.817584813151584, 
  0.575808191417845, -0.817584813151583, -0.575808191417845, 
  -0.575808191417845, 0.817584813151584, 0.575808191417845, 
  -0.817584813151584, 0.1709618887603014, 0.985277642388941, 
  -0.1709618887603012, -0.985277642388941, -0.985277642388941, 
  0.1709618887603012, 0.985277642388941, -0.1709618887603012, 
  0.992479534598710, 0.1224106751992162, -0.992479534598710, 
  -0.1224106751992164, -0.1224106751992162, 0.992479534598710, 
  0.1224106751992163, -0.992479534598710, 0.615231590580627, 
  0.788346427626606, -0.615231590580627, -0.788346427626606, 
  -0.788346427626606, 0.615231590580627, 0.788346427626606, 
  -0.615231590580627, 0.870086991108711, 0.492898192229784, 
  -0.870086991108711, -0.492898192229784, -0.492898192229784, 
  0.870086991108711, 0.492898192229784, -0.870086991108711, 
  0.266712757474898, 0.963776065795440, -0.266712757474898, 
  -0.963776065795440, -0.963776065795440, 0.266712757474898, 
  0.963776065795440, -0.266712757474898, 0.949528180593037, 
  0.313681740398892, -0.949528180593037, -0.313681740398891, 
  -0.313681740398891, 0.949528180593037, 0.313681740398892, 
  -0.949528180593037, 0.449611329654607, 0.893224301195515, 
  -0.449611329654607, -0.893224301195515, -0.893224301195515, 
  0.449611329654607, 0.893224301195515, -0.449611329654607, 
  0.757208846506485, 0.653172842953777, -0.757208846506485, 
  -0.653172842953777, -0.653172842953777, 0.757208846506485, 
  0.653172842953777, -0.757208846506485, 0.07356456359966746, 
  0.997290456678690, -0.07356456359966735, -0.997290456678690, 
  -0.997290456678690, 0.07356456359966775, 0.997290456678690, 
  -0.07356456359966744, 0.997290456678690, 0.07356456359966744, 
  -0.997290456678690, -0.07356456359966775, -0.07356456359966735, 
  0.997290456678690, 0.07356456359966746, -0.997290456678690, 
  0.653172842953777, 0.757208846506485, -0.653172842953777, 
  -0.757208846506485, -0.757208846506485, 0.653172842953777, 
  0.757208846506485, -0.653172842953777, 0.893224301195515, 
  0.449611329654607, -0.893224301195515, -0.449611329654607, 
  -0.449611329654607, 0.893224301195515, 0.449611329654607, 
  -0.893224301195515, 0.313681740398892, 0.949528180593037, 
  -0.313681740398891, -0.949528180593037, -0.949528180593037, 
  0.313681740398891, 0.949528180593037, -0.313681740398892, 
  0.963776065795440, 0.266712757474898, -0.963776065795440, 
  -0.266712757474898, -0.266712757474898, 0.963776065795440, 
  0.266712757474898, -0.963776065795440, 0.492898192229784, 
  0.870086991108711, -0.492898192229784, -0.870086991108711, 
  -0.870086991108711, 0.492898192229784, 0.870086991108711, 
  -0.492898192229784, 0.788346427626606, 0.615231590580627, 
  -0.788346427626606, -0.615231590580627, -0.615231590580627, 
  0.788346427626606, 0.615231590580627, -0.788346427626606, 
  0.1224106751992163, 0.992479534598710, -0.1224106751992162, 
  -0.992479534598710, -0.992479534598710, 0.1224106751992164, 
  0.992479534598710, -0.1224106751992162, 0.985277642388941, 
  0.1709618887603012, -0.985277642388941, -0.1709618887603012, 
  -0.1709618887603012, 0.985277642388941, 0.1709618887603014, 
  -0.985277642388941, 0.575808191417845, 0.817584813151584, 
  -0.575808191417845, -0.817584813151584, -0.817584813151583, 
  0.575808191417845, 0.817584813151584, -0.575808191417845, 
  0.844853565249707, 0.534997619887097, -0.844853565249707, 
  -0.534997619887097, -0.534997619887097, 0.844853565249707, 
  0.534997619887097, -0.844853565249707, 0.2191012401568698, 
  0.975702130038529, -0.2191012401568697, -0.975702130038529, 
  -0.975702130038528, 0.21910124015687, 0.975702130038529, 
  -0.2191012401568698, 0.932992798834739, 0.359895036534988, 
  -0.932992798834739, -0.359895036534988, -0.359895036534988, 
  0.932992798834739, 0.359895036534988, -0.932992798834739, 
  0.405241314004990, 0.914209755703531, -0.405241314004990, 
  -0.914209755703531, -0.914209755703531, 0.405241314004990, 
  0.914209755703531, -0.405241314004990, 0.724247082951467, 
  0.689540544737067, -0.724247082951467, -0.689540544737067, 
  -0.689540544737067, 0.724247082951467, 0.689540544737067, 
  -0.724247082951467, 0.02454122852291227, 0.999698818696204, 
  -0.02454122852291214, -0.999698818696204, -0.999698818696204, 
  0.02454122852291233, 0.999698818696204, -0.02454122852291229, 
  0.999924701839145, 0.01227153828571992, -0.999924701839145, 
  -0.01227153828572001, -0.01227153828571982, 0.999924701839145, 
  0.01227153828571994, -0.999924701839145, 0.698376249408973, 
  0.715730825283819, -0.698376249408973, -0.715730825283819, 
  -0.715730825283818, 0.698376249408973, 0.715730825283819, 
  -0.698376249408973, 0.919113851690058, 0.393992040061048, 
  -0.919113851690058, -0.393992040061048, -0.393992040061048, 
  0.919113851690058, 0.393992040061048, -0.919113851690058, 
  0.371317193951838, 0.928506080473215, -0.371317193951838, 
  -0.928506080473216, -0.928506080473215, 0.371317193951838, 
  0.928506080473215, -0.371317193951838, 0.978317370719628, 
  0.2071113761922186, -0.978317370719628, -0.2071113761922188, 
  -0.2071113761922184, 0.978317370719628, 0.2071113761922186, 
  -0.978317370719628, 0.545324988422047, 0.838224705554838, 
  -0.545324988422046, -0.838224705554838, -0.838224705554838, 
  0.545324988422046, 0.838224705554838, -0.545324988422046, 
  0.824589302785025, 0.565731810783613, -0.824589302785025, 
  -0.565731810783613, -0.565731810783613, 0.824589302785025, 
  0.565731810783613, -0.824589302785025, 0.183039887955141, 
  0.983105487431216, -0.1830398879551409, -0.983105487431216, 
  -0.983105487431216, 0.1830398879551409, 0.983105487431216, 
  -0.1830398879551409, 0.993906970002356, 0.1102222072938831, 
  -0.993906970002356, -0.1102222072938833, -0.1102222072938831, 
  0.993906970002356, 0.1102222072938832, -0.993906970002356, 
  0.624859488142386, 0.780737228572094, -0.624859488142386, 
  -0.780737228572095, -0.780737228572094, 0.624859488142386, 
  0.780737228572094, -0.624859488142386, 0.876070094195406, 
  0.482183772079123, -0.876070094195406, -0.482183772079123, 
  -0.482183772079123, 0.876070094195407, 0.482183772079123, 
  -0.876070094195406, 0.278519689385053, 0.960430519415566, 
  -0.278519689385053, -0.960430519415566, -0.960430519415566, 
  0.278519689385053, 0.960430519415566, -0.278519689385053, 
  0.953306040354194, 0.302005949319228, -0.953306040354194, 
  -0.302005949319228, -0.302005949319228, 0.953306040354194, 
  0.302005949319228, -0.953306040354194, 0.460538710958240, 
  0.887639620402854, -0.460538710958240, -0.887639620402854, 
  -0.887639620402854, 0.460538710958240, 0.887639620402854, 
  -0.460538710958240, 0.765167265622459, 0.643831542889791, 
  -0.765167265622459, -0.643831542889791, -0.643831542889791, 
  0.765167265622459, 0.643831542889791, -0.765167265622459, 
  0.0857973123444399, 0.996312612182778, -0.0857973123444398, 
  -0.996312612182778, -0.996312612182778, 0.0857973123444402, 
  0.996312612182778, -0.0857973123444399, 0.998118112900149, 
  0.06132073630220858, -0.998118112900149, -0.06132073630220849, 
  -0.06132073630220854, 0.998118112900149, 0.06132073630220866, 
  -0.998118112900149, 0.662415777590172, 0.749136394523459, 
  -0.662415777590172, -0.749136394523459, -0.749136394523459, 
  0.662415777590172, 0.749136394523459, -0.662415777590172, 
  0.898674465693954, 0.438616238538528, -0.898674465693954, 
  -0.438616238538528, -0.438616238538527, 0.898674465693954, 
  0.438616238538528, -0.898674465693954, 0.325310292162263, 
  0.945607325380521, -0.325310292162263, -0.945607325380521, 
  -0.945607325380521, 0.325310292162263, 0.945607325380521, 
  -0.325310292162263, 0.966976471044852, 0.254865659604515, 
  -0.966976471044852, -0.254865659604515, -0.254865659604515, 
  0.966976471044852, 0.254865659604515, -0.966976471044852, 
  0.503538383725718, 0.863972856121587, -0.503538383725717, 
  -0.863972856121587, -0.863972856121587, 0.503538383725718, 
  0.863972856121587, -0.503538383725718, 0.795836904608884, 
  0.605511041404326, -0.795836904608883, -0.605511041404326, 
  -0.605511041404325, 0.795836904608884, 0.605511041404326, 
  -0.795836904608884, 0.1345807085071262, 0.990902635427780, 
  -0.1345807085071261, -0.990902635427780, -0.990902635427780, 
  0.1345807085071263, 0.990902635427780, -0.1345807085071262, 
  0.987301418157858, 0.1588581433338614, -0.987301418157858, 
  -0.1588581433338614, -0.1588581433338613, 0.987301418157858, 
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